Linking the mesh and location of objects with different materials in different scenes
No, I don't think you can, as far as I know.

You can link objects from two scenes or even files, their positions will be kept in sync.

You can also have materials applied to Object or Object Data, so two objects can share the same object data but have different materials.

You can't however move them independently. If you use U > Make Single User > Object, you will be able to change materials but they will move them independently.

One possible workaround is to use parenting. Create an empty object, for example, and link it between the two scenes so their positions are kept in sync. Then in each scene parent a differently colored sphere to the empty.

That way the empty will guarantee they move independently, but their visual appearance will differ.